Akshually... you can’t... just the government...
https://www.hrsa.gov/vaccine-compensation/index.html
I was trying to keep it simple.
In the case of approved vaccines, you cannot formally sue the federal government. You petition a special federal court for compensation from a special fund.
If the federal court denies your petition, you may then formally sue the pharma, or the doctor who urged you to get the shot, or the clinic that gave you the shot.
Grounds for suing the pharma are usually based on damaging research or testing that they concealed from the public.
Grounds for suing a doctor, or a clinic, or your employer, are usually based on the fact that they should have informed you about publicly available information of potential harm from the vaccine.